1. Field of the Invention
This invention relates to a method for forming a calibration curve in an infrared gas analyzer for measuring the concentration of a sample gas, such as CO and CO.sub.2.
2. Description of the Prior Art
In concentration measurements for CO and CO.sub.2 contained in, for example, vehicle emission gases, non-distribution type infrared gas analyzers (NDIR) are used. Along with recent rapid developments of low-emission engines, there are ever-increasing demands for high-precision infrared gas analyzers.
One of the important factors in achieving analytical precision for the infrared gas analyzer is to properly create and maintain the calibration curve. The formation of this calibration curve has been generally carried out by using a system as shown in FIG. 3. In FIG. 3, reference number 1 represents a gas analyzing section which is constituted by, for example, an infrared gas analyzer 2 for measuring gases such as CO.sub.2, a preamplifier 3 for appropriately waveform-shaping the output of the infrared gas analyzer 2, and an AD converter 4 for A/D converting the output of the infrared gas analyzer that is inputted thereto through the preamplifier 3.
Reference numeral 5 represents a gas divider for generating some calibration gas SG having a known concentration and for supplying gas SG to the above-mentioned gas analyzer 1. This is arranged so that a component gas (span gas) CG of which the concentration is preliminarily determined and a dilute gas (for example, nitrogen gas or air) DG which dilutes the component gas to a predetermined concentration are given as inputs, and the generated gas SG is outputted therefrom.
Reference numeral 6 is an Main Control Unit (MCU) for supervising and controlling the entire system, which MCU is constituted by a computer. The MCU 6 controls the gas analyzing section 1 and the gas divider 5 through interface controllers (IFC) 7 and 8, and also has a computing function for carrying out concentration calculations based upon the output from the gas analyzer 1. Reference numeral 9 is a display device, for example, a touch panel display, connected to the MCU 6. Moreover, reference numeral 10 is a storage medium, for example, a hard disk, etc., which stores measured data and is provided with a plurality of files containing data to be displayed on the device 9.
In the above-mentioned arrangement, the calibration-use component gas CG and the dilute gas DG are supplied to the gas divider 5, and the component gas SG having an appropriate concentration is obtained. Then, the component gas SG, adjusted to the appropriate concentration, is supplied to the infrared gas analyzer 2 in the gas analyzing section 1, and output signals (measured values) from the infrared gas analyzer 2 are obtained as a plurality of points (not less than four as measured points). Then, these are inputted to the MCU 6 so that, based upon not less than four measured values thus obtained, the calibration curve is approximated by a polynomial not greater than its 4th order by using the method of least squares
In other words, the calibration curve f(x) is represented as follows: EQU f(x)=a.sub.0 +a.sub.1 x+a.sub.2 x.sup.2 +a.sub.3 x.sup.3 +a.sub.4 x.sup.4,
Assuming that the measured value is (x.sub.i, y.sub.i), coefficients, a.sub.0 through a.sub.4 are obtained from the following equations: EQU .DELTA..epsilon..sub.i.sup.2 ={y.sub.i -(a.sub.0 +a.sub.1 x.sub.i +a.sub.2 x.sub.i.sup.2 +a.sub.3 x.sub.i.sup.3 +a.sub.4 x.sub.i.sup.4)}.sup.2 (1) EQU .delta..SIGMA..epsilon..sub.i.sup.2 /.differential.aj=0(j=0-4) (2)
Here, it is well known in the art that the principle of measurement of the infrared gas analyzer 2 follows the Lambert-Beer's law as shown in the following equation (3). EQU I=I.sub.0 exp(-.mu.cL) (3)
where I.sub.0 : intensity of incident light, I: intensity of transmitted light, .mu.: absorption coefficient, c: concentration of a sample gas to be measured, L: thickness of gas layer.
The above-mentioned equation (3) indicates that the transmittance, which is the rate of change in the intensity of transmitted light, can be approximated to a first-order equation in a low-density area, and that as the concentration increases, it can be explicated to a higher order equation.
For example, in the exhaust-gas measurements of vehicles, it is necessary to approximate the calibration curve used in the infrared gas analyzer 2 by using a polynomial not greater than its 4th order; and in the case when the measuring range is wide, that is, in the case of a combined area of the area that can be approximated by a first-order equation and the area that can be approximated by a higher order (2nd, 3rd, 4th) equation, the precision of the calibration curve in the area that can be approximated by a first-order equation tends to deteriorate with systematic error.
FIG. 4 explains the precision of the above-mentioned calibration curve, and in FIG. 4, curve 41, indicated by a solid line, is a calibration curve, and a curve indicated by a phantom line indicates the concentration obtained by actual measurements. Moreover, symbol I is an area that can be approximated by a first-order equation, and II is an area that can be approximated by a high order (2nd, 3rd, 4th) equation. As shown in FIG. 4, the calibration curve 41 is approximately coincident with the measurement curve 42 within the area 41 that can be approximated by a higher order equation, which corresponds to a high concentration area; however, in the first-order-equation approximation area which corresponds to a low-concentration area, it deviates from the curve to a great degree, with the result that the precision of the calibration curve is lowered and E.sub.rr % (error %) in the low-concentration area sometimes fails to meet standard criteria (.+-.2.0% PT). Here, % PT refers to the error in reading a value. The error with the calibration curve 41 with respect to the measurement curve 42 in the above-mentioned infrared gas analyzer is classified as the system error that inevitably occurs in the measuring principle of the infrared gas analyzer.